Given a Tonelli Hamiltonian on a compact manifold, we can construct a compact invariant subset of the cotangent bundle enjoying variational properties which has the distinguished property of being a Lipschitz graph. This set called Aubry set captures many important features of the Hamiltonian dynamics. Fathi established a bridge between the Aubry-Mather theory and the properties of viscosity solutions and subsolutions of the critical Hamilton-Jacobi equation, thus giving rise to the weak KAM theory. A famous open problem concerning the structure of the Aubry set is the so-called Mañé conjecture, which states that, for a generic Hamiltonian, the Aubry set is a hyperbolic orbit. The aim of this talk is to present some of the ideas to prove that generic Aubry sets on surfaces are hyperbolic. This is a joint work with Alessio Figalli and Gonzalo Contreras.